maximal interval of existence of ordinary differential equations

Let $E\subset W$ where $W$ is a normed vector space, $f\in C^1(E)$ is a continuous differentiable map $f:E\to W$. Furthermore consider the ordinary differential equation

$$\dot{x}=f(x)$$

with the initial condition

$x(0)=x_0$.

For all $x_0\in E$ there exists a unique solution

$$x:I\to E$$

where $I=[-a,a]$, which also satify the initial condition of the initial value problem. Then there exists a maximal interval of existence $J=(\alpha ,\beta )$ such that $I\subset J$ and there exists a unique solution

$x:J\to E$.